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The modern theory of Partial Differential Equations relies heavily on functional analytic methods. With the advent of high speed computers, numerical methods, like the finite element method, have revolutionized areas like computational fluid dynamics and structural analysis. These rely on the study of weak solutions to PDEs and functional analysis plays a dominant role in this. At the core of the ideas involved lie the theory of distributions and the important function spaces, called the Sobolev spaces. These spaces form a natural framework in which we study generalized (i.e. weak) solutions of boundary value problems. This course will develop, in detail, the theory of distributions and study the important properties of Sobolev spaces. These will be applied to the study of weak solutions of elliptic boundary value problems. The theory of semigroups of operators on a Banach space will also be developped. This, together with Sobolev spaces will help us to study evolution equations. Prerequisites for this course are analysis, topology, linear algebra, functional analysis and measure theory (especially, the theory of L-p spaces).INTENDED AUDIENCE : MSc (Mathematics) and abovePRE-REQUISITES : MSc Real Analysis, Topology, Linear Algebra,Measure Theory, Functional Analysis